Traveling waves of a Keller-Segel system with bounded N-shaped chemical degradation and small diffusion

The existence of the traveling pulse-front solutions of a Keller-Segel model with bounded N-shaped degradation of the chemical signal is studied. We consider this model with a singularly perturbed term in the diffusion rates. We obtain the singular orbits by finding the maximal invariant set and LaSalle's invariance principle. The singular orbits are then shown to persist for sufficiently small diffusion by geometric singular perturbation theory, together with continuous dependence and the local dynamics near the terminal equilibrium. Our results show that traveling pulse-front solutions with monotone or oscillatory tails both exist, which may connect a saddle to a stable node or a stable focus.